The document discusses energy minimization in the context of bioinformatics, detailing the process of finding atomic arrangements that minimize forces based on potential energy surfaces (PES). It outlines key concepts such as the significance of geometry optimization, mathematical interpretations of atomic positions and energy, and various optimization algorithms, specifically steepest descent and conjugate gradient methods. The emphasis is on how optimized molecular structures can be used for various experimental and theoretical investigations in chemistry.

1. Energy Minimization
Aishwarya M. Rane
M.Sc. Bioinformatics(part-2)
G. N. Khalsa CollegeJul 24, 2018 1

2. Contents
Energy Minimization
1. Introduction
2. Potential energy surface
3. Basicsof energy minimization
Energy Minimization Algorithms
1. Steepest Descent
2. ConjugateGradient
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3. Introduction
• energy minimization is the process of finding an arrangement
in space of a collection of atoms where, according to some
computational model of chemical bonding, the net inter-
atomic force on each atom is acceptably close to zero and the
position on the potential energy surface (PES) is a stationary
point.
single molecule, an ion,
a transition state or even a
collection of any of these
Quantum mechanics
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4. General Terms
• A potential energy surface (PES) describes the
energy of a system, especially a collection of atoms, in
terms of certain parameters, normally the positions of
theatoms.
Explores properties of structures
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5. Forexample
•When optimizing the geometry
of a water molecule, our aim is
to obtain thehydrogen-oxygen
bond lengths and thehydrogen-
oxygen-hydrogen bond angle
which minimizetheforcesthat
would otherwisebepulling atoms
together or pushing them apart.
Fig1: PES for water molecule: Shows the energy minimum
corresponding to optimized molecular structure for water- O-H bond
length of 0.0958nm and H-O-H bond angle of 104.5°
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6. Significance of geometry
optimization
• Physically significant structures.
• Optimized structures often correspond to a substance as it is
found in nature.
• Optimized structurescan beused in avariety of
experimental and theoretical investigationsin thefieldsof
chemical structure, thermodynamics,chemicalkinetics.
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7. Molecular Geometry and
Mathematical Interpretation
Thegeometry of aset of atoms
Vector of theatoms' positions
Described byDescribed by
Single
atom
Cartesian
Coordinates
Cartesian
Coordinates
Molecule
Internal
Coordinates
Internal
Coordinates
Bond lengths , Bond
Angles , Dihedral Angles
Bond lengths , Bond
Angles , Dihedral Angles
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8. Mathematical Interpretation
• r– A vectorthe describes the atoms' positions
• E(r) – Energy as a function of position
• Mathematically, our aim is to to find the
value of r for which E(r) is at a local
minimum.
Smallest value of a function
Obtained by calculating
∂E/∂r
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9. Mathematical Interpretation
• Second aim is to find the curvature of the
PES at r.
Obtained by calculating
∂∂E/∂ri∂rj
Hessian matrixHessian matrixJul 24, 2018 9

10.  Thecomputational model that providesan approximate E(r) could bebased
on
1. Quantum mechanics
2. Forcefields
3. Or acombination of thosein caseof QM/MM.
 Using thiscomputational model and an initial information of thecorrect
geometry, an iterativeoptimization procedureisfollowed, for example:
 Calculate the force on each atom (that is, -∂ E/∂ r).
 If the force is less than some threshold, finish.
 Otherwise, move the atoms by some computed step .
 Repeat from the start.
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11. Energy Optimization
Algorithms
• An optimization algorithm can use some or all
of E(r) , ∂E/∂r and ∂∂E/∂ri∂rj to try to minimize the
forces.
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12. Steepest Descent
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13. Steepest Descent
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Fig2 : Steepest descent
• Slow near minimum.
• Used for structures away
from minimum.
• Used asrough and
introductory method
followed by moreadvanced
methods.

14. Conjugate Gradient
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Valuefrom
previousstep

15. Conjugate Gradient
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Fig3: ConjugateGradient
• Used for larger system.
• Morecomputational power
and requirements.
• Moreefficient
convergence.

16. Jul 24, 2018 16